Stability theorems for semi–groups via Multiplier theorems

Lutz Weis, Karlsruhe

1 Introduction

A classical theorem of Liapunov states that if all eigenvalues λ of a matrix A have Re λ < 0, then all solutions of the Cauchy problem

y′(t) = Ay(t), y(0) = y_{0}

are exponentiell stable, i. e. there are ε > 0 and M < ∞ such that y(t) = e^{tA}y_{0} satisfies

∥y(t)∥ ≤ Me^{−εt}.  (1.1)

In recent years, an analoguous stability theory was developped for c_{0}–semigroups on infinite dimensional Banach spaces. If A is the generator of a c_{0}–semigroup (T_{t})_{t≥0} on a Banach space X, we denote by

s(A) = sup{Re λ : λ ∈ σ(A)}  and
s_{0}(A) = inf{ω : sup_{Reλ≥ω} ∥R(λ, A)∥ < ∞}

(1.2)

the spectral bound of A and the boundedness abscissa of R(λ, A) respectively. For every x ∈ X,

w(x) = inf{ω : sup_{t≥0} ∥e^{−ωt}T_{t}x∥ < ∞}  (1.3)

is the growth bound of the mild solution y(t) = T_{t}x of

y′(t) = Ay(t),  y(0) = x.  (1.4)

It was shown in [17] and also in [8], [9], that

w(x) ≤ s_{0}(A) for all x ∈ D(A),  (1.5)

i. e. if s_{∞}(A) < 0 and y(t) is a classical solution of (4) (because of x ∈ D(A)), then y(t) is exponentially stable. Since for a matrix A on X = C^{n} we have

s_{∞}(A) = s(A) = sup{Re λ : λ eigenvalue of A}

and D(A) = X, (5) seems to be the proper generalization of Liapunov’s theorem to the infinite dimensional setting, as far as classical solutions of (4) are concerned. For mild solutions y(t) = T_{t}x of (4) with x ∈ X one has

ω(x) ≤ s(A) = s_{∞}(A) for all x ∈ X  (1.6)

only in special circumstances. It is well known ([10]) that (6) holds if (T_{t}) is an analytic semigroup, but more recently it was shown in [15] (see also [9], [16]) that (6) also holds for positive semigroups on L_{p}–spaces. If X has additional geometric properties, e. g. if X is uniformly convex or a Banach space of Fourier–type 1 < p ≤ 2, then (cf. [17]) one obtains a result in between (1.5) and (1.6): w(x) ≤ s_{0}(A) for all x in the domain of a fractional power (−A)^{α}, where 0 < α < 1 depends on the geometry of X. In this note we give alternative proofs for these results based on Multiplier theorems for operator–valued functions M(λ) ∈ B(X), λ ∈ IR, on vector– valued Besov spaces B_{u,v}^{s}(IR^{n}, X). These theorems from [4] and discovered independently by Amann [1] assure that

K̂f(λ) = M(λ)(f̂(λ)), λ ∈ IR^{n}  (1.7)

defines a bounded operator K on B_{u,v}^{s}(X), if M satisfies various conditions of Mihlin type, e. g.

∥(1 + |λ|^{α})D^{α}M(λ)∥ ≤ C for λ ∈ IR^{n}, |α| ≤ n + 1.  (1.8)

Here f̂ denotes the Fourier–transform

f̂(t) = ∫_{IR^{n}} e^{−it·s}f(s)ds.  (1.9)

If X has Fourier–type p, p ≤ 2, i. e. the vector–valued Hausdorff–Young inequality

∥f̂∥_{L_{p′}(IR^{n},X)} ≤ C∥f∥_{L_{p}(IR^{n},X)}  (1.10)

holds, it is enough to assume that

M ∈ B_{p,1}^{1/p}(IR^{n})  (1.11)

These multiplier theorems are stated preciscely in \underline{Section 2} where we also give a short sketch of their proof. In \underline{Section 3} we explain how stability theorems can be reduced to multiplier theorems and we give a new proof of (1.5) and its improvements for Banach spaces of Fourier–type p. An important consequence of this proof and examples in [17] will be, that condition (9) is optimal for Banach spaces of Fourier–type p. We also include a simple proof for the convexity of the growth function α → ω((−A)^{−α}x) from [17] (see 3.5. for details), which gives a partial answer to a question of van Neerven concerning individual stability estimates. Finally, in \underline{Section 4} we sketch an alternative proof to the short proof of (6) for positive semigroups on L_{p} that will appear in [16]. A final example gives a negative answer to a question of van Neerven concerning the growth bound of non–positive semigroups on L_{p}(Ω).

2 Operator–valued Fourier Multipliers on Vector–valued Besov spaces

It was observed by G. Pisier that if an operator–valued function t ∈ IR^{n} → M(t) ∈ B(X) satisfying the Mihlin condition (1.8) defines a bounded operator K on L_{p}(IR^{n}, X), 1 < p < ∞, by

K̂f(λ) = M(λ)(f̂(λ))

then X is necessarily a Hilbert space. To obtain multiplier results for arbitrary Banach spaces, we are therefore lead to replace the Bochner spaces L_{p}(IR^{n}, X) by the Besov spaces B_{u,v}^{s}(IR^{n}, X). Let us recall their definition. Put

I_{0} = {x ∈ IR^{n} : |x| ≤ 2}, I_{j} = {x ∈ IR^{n} : 2^{j−1} ≤ |x| ≤ 2^{j+1}}, j ∈ IN.

We fix a sequence ϕ_{j} ∈ C^{∞}(IR^{n}), j = 0, 1, 2, …, such that

suppϕ_{j} ⊂ I_{j},  ∑_{j=0}^{∞} ϕ_{j}(s) = 1 on IR^{n},  sup_{j} ∫_{IR^{n}} |ϕ̂_{j}(t)|dt < ∞. (2.1)

Then B_{u,v}^{s}(IR^{n}, X), s ∈ IR, 1 ≤ u, v ≤ ∞, consists of all X–valued distributions f for which the Besov norm

∥f∥_{B_{u,v}^{s}} = (∑_{j=0}^{∞}(2^{js}∥ϕ̂_{j} ∗ f∥_{u})^{v})^{1/v}  (2.2)

is finite. (Different choices of sequence ϕ_{j} lead to equivalent norms; for this and other properties of these Besov spaces see e. g. [12], [13].) A well–known equivalent norm on B_{u,v}^{s}(IR, X) for 0 < s < 1 is

∥|f∥|_{B_{u,v}^{s}} = ∥f∥_{L_{u}} + (∫ ∫ ||\frac{f(t) − f(r)}{|t − r|^{s}} ||_{L_{u}}^{v} |t − r|^{−1}dt dr)^{1/v} .

B_{u,v}^{s}(IR_{+}, X) is the space of distribution in B_{u,v}^{s}(IR, X) with support in IR_{+}. Our results also depend on the geometry of the underlying Banach space X. We say that X has \underline{Fourier–type p}, 1 ≤ p ≤ 2, if the Fourier–transform (1.9) extends to a bounded operator

f ∈ L_{p}(IR, X) → f̂ ∈ L_{p′}(IR, X),  \frac{1}{p} + \frac{1}{p′} = 1,  (2.3)

(cf.[11]). Obviously, every Banach space has Fourier–type 1. On the other hand, a Banach space has Fourier–type 2 if and only if X is a Hilbert space. A space L_{q}(Ω, μ) has Fourier–type p = min(q, \frac{q}{q−1}) and Bourgain has shown in [3], that every B–convex (in particular every uniformly convex) Banach space has type p for some p > 1. Also the dual space, every closed subspace and quotient space of a Banach space X has the same Fourier–type as X. Finally, we fix two functions ψ_{0}, ψ_{1} ∈ C^{∞}(IR^{n}) with

supp ψ_{0} ⊂ {x ∈ IR^{n} : |x| ≤ 4}, ψ_{0}(x) = 1 for |x| ≤ 2
supp ψ_{1} ⊂ \{x ∈ IR^{n} : \frac{1}{4} ≤ |x| ≤ 4\}, ψ_{0}(x) = 1 for \frac{1}{2} ≤ |x| ≤ 2 (2.4)

In [4] we prove the following Multiplier theorem:

2.1. Theorem: Let X and Y have Fourier–type p, 1 ≤ p ≤ 2. Assume that the function t ∈ IR^{n} → M(t) ∈ B(X, Y) satisfies:

i) ∥ψ_{1}(·)M(2^{k}·)x∥_{B_{p,1}^{n/p}(Y)} ≤ C∥x∥ for all x ∈ X, k ∈ IN

∥ψ_{0}(·)M(·)x∥_{B_{p,1}^{n/p}(Y)} ≤ C∥x∥ for all x ∈ X.

ii) ∥ψ_{1}(·)M^{∗}(2^{k}·)y^{∗}∥_{B_{p,1}^{n/p}(X^{∗})} ≤ C∥y^{∗}∥ for all y^{∗} ∈ Y^{∗}, k ∈ IN

∥ψ_{0}(·)M^{∗}(·)y^{∗}∥_{B_{p,1}^{n/p}(X^{∗})} ≤ C∥y^{∗}∥ for all y^{∗} ∈ Y^{∗}

Then K̂f(λ) = M(λ)[f̂(λ)], λ ∈ IR^{n}, defines a bounded operator

K : B_{u,v}^{s}(IR^{n}, X) → B_{u,v}^{s}(IR^{n}, Y).

for s ∈ IR, 1 ≤ u, v ≤ ∞.

2.2. Remark: a) It is checked in [4] that i) and ii) are in particular fulfilled if M satisfies the Mihlin condition

∥(1 + |λ|)^{|α|}D^{α}M(λ)∥ ≤ C for λ ∈ IR^{n}, |α| ≤ n + 1.  (2.5)

This case was proved independently in [1] by Amann. If X has Fourier–type p, it is enough to have (2.5) for |α| ≤ [\frac{n}{p}] + 1. In particular, if X is uniformly convex it is enough to consider derivatives of order |α| ≤ n. Alternatively, it is enough to assume

∥D^{α}M(λ)∥ ≤ C(1 + |λ|)^{−n}  (2.6)

for all |α| ≤ n + 1 (or |α| ≤ \frac{n}{p} + 1 if X and Y have Fourier–type p).

b) Cf. [4] one can replace the assumptions i) and ii) by

i’) ∥M(·)x∥_{W_{p}^{r}(I_{j},Y)} ≤ C∥x∥_{X} for all x ∈ X, j = 0, 1, 2, …

ii’) ∥M^{∗}(·)y^{∗}∥_{W_{p}^{r}(I_{j},X^{∗})} ≤ C∥y^{∗}∥_{Y^{∗}} for all y^{∗} ∈ Y^{∗}, j = 0, 1, 2 …

for r ∈ IN, r > n/p.

\underline{2.3. The proof of theorem 2.1.:} We need

1) an estimate for the Fourier–transform (theorem 3.1. in [4]), namely

∥ĝ∥_{L_{1}(IR^{n},X)} ≤ C∥g∥_{B_{p,1}^{n/p}}(IR^{n}, X)

2) the following extention of Youngs inequality (Lemma 3.2. in [4]): If k : IR^{N} → B(X, Y) satisfies

∫_{IR^{n}} ∥k(s)x∥_{Y} ds ≤ C∥x∥_{X}, ∫_{IR^{n}} ∥k(s)^{∗}y^{∗}∥_{X^{∗}}ds ≤ C∥y^{∗}∥_{Y^{∗}}

then T f(t) = ∫_{IR^{n}} k(t − s)f(s)ds defines a bounded operator

T : L_{p}(IR^{n}, X) → L_{p}(IR^{n}, Y).

Now observe that K restricted to the ”blocks” φ_{j}∗f ∈ L_{p}(IR^{n}, X) of B_{u,v}^{s}(IR^{n}, X) is essentially the convolution with k_{j} = [ψ(2^{−j}·)M(·)]^{∨}. Applying 1) to g = ψ(2^{−j}M(·)) we obtain that k_{j} satisfies the assumptions of 2) and, by the definition of the norm (2.2), we obtain the theorem by estimating the ”blocks” separately (and uniformly)

∥φ_{j} ∗ Kf∥ ≤ ∥k_{j} ∗ (φ_{j} ∗ f)∥ ≤ C∥φ_{j} ∗ f∥

(see [4] for details).

3 Stability of solutions with initial values in D((−A)^{α})

Let (T_{t})_{t≥0} be a c_{0}–semigroup on X with ∥T_{t}∥ ≤ Me^{ωt} and s_{0}(A) < 0. Let X_{α} be the domain of the fractional power (a − A)^{α}, where α ≥ 0 and a > ω (see e. g. [14]) with the graph norm. In particular, we have for all w > ω and x ∈ X_{α} that

∫_{0}^{∞}∥e^{−wt}T_{t}x∥_{X_{α}}^{p}dt < ∞

The basic idea in applying multiplier theorems to stability results is now the following: Observe that

∫_{0}^{t} T_{t−s}(e^{−ws}T_{s}x)ds = \frac{1}{α}(1 − e^{−wt})T_{t}x  (3.1)

If we could assure, that the operator

Kf(t) = ∫_{0}^{t} T_{t−s}(f(s))ds = (T_{t}) ⋆ f  (3.2)

maps L_{p}(IR_{+}, X_{α}) into L_{p}(IR_{+}, X) by applying a multiplier result to

M(λ) = ” ∫_{0}^{∞}e^{−iλt}T_{t}dt” = R(iλ, A),  λ ∈ IR  (3.3)

(where the ”integral” should be understood in the distributional sense), then t → \frac{1}{α}(1 − e^{−wt})T_{t}x would belong to L_{p}(IR_{+}, X). Hence

∫_{0}^{∞}∥T_{t}x∥_{X}^{p}dt < ∞

and a result of Pazy ([10] Sect. 4.4) would tell us that

∥T_{t}x∥_{X} → 0 for t → ∞.

The proofs below will follow this outline but they will be somewhat complicated by the fact that our Multiplier Theorems do not garanty that K is bounded in Bochner spaces but only in Besov spaces. With the notation of the introduction we prove first

3.1. Theorem: Let (T_{t}) be a c_{0}–semigroup on X. Then

w(x) ≤ s_{0}(A) for all x ∈ D(A).

\underline{Proof:} Let s_{0}(A) < ∞ and ∥T_{t}∥ ≤ Me^{ωt}. Then for x ∈ Y = D(A)

∥R(iλ, A)x∥ ≤ \frac{C}{|λ|}∥x∥_{Y}

by Lemma 3.3. Also

∥\frac{d}{dx}R(iλ, A)x∥_{X} = ∥R(iλ, A)^{2}x∥_{X} ≤ ∥R(iλ, A)∥_{B(X)}∥R(iλ, A)x∥ ≤ \frac{C^{2}}{|λ|} ∥x∥_{Y} .

Hence

||R(iλ, A)||_{Y →X} ≤ \frac{C}{|λ|},  ||\frac{d}{dλ}R(iλ, A)||_{Y →X} ≤ \frac{C^{2}}{|λ|}  (3.4)

and in the same way we can show that ||\frac{d^{2}}{dλ^{2}} R(iλ, A)||_{Y →X} ≤ \frac{C^{3}}{|λ|} .

By remark 2.2.a) M(λ) = R(iλ, A) ∈ B(Y, X) satisfies the assumptions of Theorem 2.1. For f ∈ C_{c}(IR_{+}) and w > ω we have that

∫_{0}^{∞}e^{−iλt} [∫_{0}^{t} T_{s}f(t − s)ds] dt = ∫_{0}^{∞}e^{−i(λ−w)t} [∫_{0}^{t} e^{−ws}T_{s}e^{−w(t−s)}f(t − s)ds] dt
= R̃(λ − w, A)[f̃(λ − w)] = R(λ, A)[f̂(λ)] = M(λ)[f̂(λ)]

where

R̃(λ)x = ∫ e^{−λt}[e^{−wt}T_{t}x]dt = R(λ + w, A)
f̃(λ) = ∫ e^{−λt}[e^{−wt}f(t)]dt = f̂(λ + w).

Hence Theorem 1.2 implies that for all ε > 0

K : B_{u,v}^{ε}(IR_{+}, Y) → B_{u,v}^{ε}(IR_{+}, X)  (3.5)

If x ∈ X_{1+2ε}, then by Lemma 3.4 we have for w > ω that

(t → f(t) = e^{−wt}T_{t}x) ∈ B_{1,u}^{ε}(IR_{+}, X).

By (3.1) and (3.5) it follows that

(t → \frac{1}{α}(1 − e^{−wt})T_{t}x) ∈ B_{1,u}^{ε}(IR_{+}, X) ⊂ L_{1}(IR_{+}, X).

Hence ∫_{0}^{∞}∥T_{t}x∥_{X}dt < ∞ and by Pazy’s result we have ∥T_{t}x∥ → 0 for t → ∞.

This shows that

ω_{α}(A) := sup{ω(x) : x ∈ X_{α}} ≤ 0

for all α > 1. Since the function α ∈ (0, ∞) → ω_{α}(A) ∈ IR is continuous by Lemma 3.5 it follows that ω_{1}(A) ≤ 0 and the proof is complete.

If we make additional assumptions on the geometry of X Theorem 3.1. can be improved as follows:

3.2. Theorem Let X be a Banach space of Fourier–type p with 1 < p ≤ 2 and let (T_{t})_{t≥0} be a c_{0}–semigroup on X. Then

w(x) ≤ s_{0}(A) for all x ∈ X_{\frac{1}{p} − \frac{1}{p′}} .

\underline{Remark:} To illustrate this result, consider the scale of spaces X = L_{q}(Ω, μ). For the Hilbert space case q = 2 we get the best possible result: ω(x) ≤ s_{0}(A) for all x ∈ X. (This is indeed Gearhart’s result.) For q = 1 we have the worst result: ω(x) ≤ s_{0}(A) is only garantied for x ∈ D(A). For 1 < q < 2 the result improves as we get closer to Hilbert–space and for Banach spaces of Fourier type p it was shown in [17] that x ∈ X_{\frac{1}{p} − \frac{1}{p′}} is indeed best possible (see also Example 4.4 below).

\underline{Proof:} Let s_{0}(A) < 0 and ∥T_{t}∥ ≤ Me^{ωt}. Put \frac{1}{r} = \frac{1}{p} − \frac{1}{p′} = \frac{2}{p} − 1. We will use

the following variant of (3.1):

∫_{0}^{t} sT_{s}(e^{−w(t−s)}T_{t−s}x)ds = g(t)T_{t}x  (3.6)

where g(t) = t/w − \frac{1}{w^{2}} (1 − e^{−wt}) is larger than 1 for t large enough. Similarly as the proof of Theorem 3.1. one shows for f ∈ C_{c}(IR, X)

∫_{0}^{∞}e^{−iλt} [∫_{0}^{t} sT_{s}f(t − s)ds] = R(iλ, A)^{2}f̂(λ).  (3.7)

Hence

Kf(t) = ∫_{0}^{t} sT_{s}f(t − s)ds = [M(λ)f̂(λ)]^{∨}, M(λ) = R(iλ, A)^{2}

In order to apply Theorem 2.1 we want to show that

i) ∥M(·)x∥_{W_{p}^{1}(J_{k},X)} ≤ C∥x∥_{X_{\frac{1}{r}}}

ii) ∥M^{∗}(·)y^{∗}∥_{W_{p}^{1}(J_{k},X∗ ; \frac{1}{r})} ≤ C∥y^{∗}∥_{X^{∗}}

for all k ∈ IN_{0}. To check i) note that for x ∈ D(A) and α = \frac{1}{r} we get from Lemma 3.3 b) and

\frac{1}{p} = \frac{1}{r} + \frac{1}{p′}

( ∫_{2^{k−1}}^{2^{k+2}}∥R(iλ, A)^{2}x∥^{p}dλ)^{\frac{1}{p}} ≤ ( ∫_{2^{k−1}}^{2^{k+2}}|λ|^{−1}dλ)^{\frac{1}{r}} (∫ [∥R(iλ, A)^{2}x∥|λ|^{\frac{1}{r}} ]^{p′} dλ)^{\frac{1}{p′}}
≤ C∥x∥_{X_{\frac{1}{r}}}

Since M(λ)′ = −2iR(iλ, A)^{3}x and ∥R(iλ, A)^{3}x∥ ≤ C∥R(iλ, A)^{2}x∥ the same

estimate holds for \frac{d}{dλ}M(λ) and we get i). ii) follows in the same way from Lemma 3.3.c. Now Theorem 2.1 and remark 2.2.b garanties that

K : B_{u,v}^{ε}(IR_{+}, X_{\frac{1}{r}}) → B_{u,v}^{ε}(IR_{+}, X)  (3.8)

for all ε > 0. If x ∈ X_{\frac{1}{r} +2ε}, then by Lemma 3.4 we have for w > ω

(t → f(t) = e^{−wt}T_{t}x) ∈ B_{1,u}^{ε}(IR_{+}, X).

Now (3.6) and (3.8) give

(t → g(t)T_{t}x) ∈ B_{1,u}^{ε}(IR_{+}, X) ⊂ L_{1}(IR_{+}, X).

By Pazy’s result we obtain ∥T_{t}x∥ → 0 for t → ∞ and we have shown

ω_{α}(A) = sup{ω(x) : x ∈ X_{α}} ≤ 0

for α > \frac{1}{r}. By Lemma 3.5 α → ω_{α}(A) is continuous and we get ω_{\frac{1}{r}} (A) ≤ 0

\underline{3.3 Lemma:} Let A be the generator of a c_{0}–semigroup T_{t} on X with s_{0}(A) < 0.

a) If x ∈ X_{α}, 0 ≤ α ≤ 1, then for λ ∈ IR

∥R(iλ, A)x∥_{X} ≤ \frac{C}{|λ|^{α}} ∥x∥_{X_{α}}.

b) If X has Fourier–type p and x ∈ X_{α}, then

(∫ [∥R(iλ, A)^{2}x∥_{X}|λ|^{α}]^{p′}dλ)^{\frac{1}{p′}} ≤ C∥x∥_{X_{α}}.

c) If X has Fourier–type p and x^{∗} ∈ X^{∗}, then

(∫ (∥R(iλ, A^{∗})^{2}x^{∗}∥_{X_{α}^{∗}}|λ|^{α})^{p′}dλ)^{\frac{1}{p′}} ≤ C∥x^{∗}∥_{X^{∗}}.

\underline{Proof:} a) Since s_{0}(A) < ∞, we have C = sup_{λ∈IR} ∥R(iλ, A)∥ < ∞ and

∥R(iλ, A)x∥ ≤ \frac{1}{|λ|}∥R(iλ, A)(Ax) + x∥ ≤ \frac{C + 1}{|λ|}∥x∥_{D(A)}.  (3.9)

This gives the claim for α = 0 and α = 1. For 0 < α < 1 we assume first that the growth bound w(A) of (T_{t}) is smaller than 0, so that we can apply the inequality

∥(−A)^{1−α}y∥ ≤ C∥y∥^{α}∥Ay∥^{1−α}, y ∈ D(A).  (3.10)

With y = R(iλ, A)(−A)^{α−1}x we obtain from (1)

∥R(iλ, A)x∥ = ∥(−A)^{1−α}y∥ ≤ C∥R(iλ, A)(−A)^{α−1}x∥^{α}∥AR(iλ, A)(−A)^{α−1}x∥^{1−α}
≤ [\frac{C + 1}{|λ|}∥A(−A)^{α−1}x∥]^{α} [C∥A(−A)^{α−1}x∥]^{1−α}
≤ \frac{C}{|λ|^{α}} ∥(−A)^{α}x∥.

For general (T_{t}) choose w > w(A) and apply this estimate to e^{−wt}T_{t}

∥R(iλ + w, A)x∥ ≤ \frac{C}{|λ|^{α}} ∥x∥_{X_{α}}.  (3.11)

The resolvent equation can be rewritten as

R(iλ, A) = [I + wR(iλ, A)] R(w + iλ, A) so, that
∥R(iλ, A)x∥ ≤ (1 + wC)∥R(w + iλ, A)x∥.  (3.12)

Now the claim follows from (3.11) and (3.12).

b) Since X_{α} is isomorph to the graph of (−A)^{α} as a subspace of X × X, X_{α} has the same Fourier–type as X. By a)

∥R(λ, A)^{2}x∥ ≤ \frac{C}{|λ|^{α}} ∥R(iλ, A)x∥_{X_{α}}.  (3.13)

T_{t} is also a c_{0}–semigroup on X_{α} and for w larger than the growth bound of T_{t} on X_{α}, we obtain from

R(w + iλ, A)x = ∫ e^{−iλt}[e^{−wt}T_{t}x]dt

and the Fourier–type of X_{α} that

(∫ ∥R(w + iλ, A)x∥_{X}^{p′}dλ)^{\frac{1}{p′}} ≤ C (∫ ∥e^{−wt}T_{t}x∥_{X_{α}}^{p}dt)^{\frac{1}{p}} ≤ D∥x∥_{X_{α}}. (3.14)

Combining (3.13), (3.12) and (3.14) gives the claim.

c) By part a) we have

\begin{aligned}∥R(λ, A^{∗})^{2}x∥_{X_{α}^{∗}} & ≤ & ∥R(λ, A)^{∗}∥_{X^{∗}→X_{α}^{∗}}∥R(λ, A^{∗})x∥_{X}∗ \\ & ≤ & \frac{C}{|λ|^{α}} ∥R(λ, A^{∗})x^{∗}∥_{X^{∗}} \end{aligned}  (3.15)

Since X^{∗} has the same Fourier–type as X we have

(∫ ∥R(w + iλ, A^{∗})x^{∗}∥_{X_{α}^{∗}}^{p′}dλ)^{\frac{1}{p′}} ≤ C (∫ ∥e^{−wt}T_{t}^{∗}x^{∗}∥_{X^{∗}}^{p}dt)^{\frac{1}{p}} ≤ D∥x^{∗}∥_{X^{∗}}.

(3.16)

Now combine (3.15), (3.12) for A^{∗} and (3.16) to get c).

\underline{3.4. Lemma:} Let ε > 0, α > 0 and w > w(A). If x ∈ X_{β} with β > α + ε than the function t → e^{−wt}T_{t} x, t ≥ 0, belongs to B_{1,∞}^{ε}(IR_{+}, X_{α}).

\underline{Proof:} Choose γ with α < γ(1 − ε) < β − ε. The map

U : X → L_{1}(IR_{+}, X), Ux(t) = e^{−wt}T_{t}x

also defines bounded operators

X_{γ} ⟶^{U} L_{1}(IR_{+}, X_{γ}) ⊂ B_{1,∞}^{0}(IR_{+}, X_{γ})
X_{1} ⟶^{U} W_{1}^{1}(IR_{+}, X) ⊂ B_{1,∞}^{1}(IR_{+}, X).

By interpolation (see [14], [13]) we get

(X_{γ}, X_{1})_{ε} ⟶^{U} B_{1,∞}^{ε}(IR_{+}, (X_{γ}, X)_{ε}).

Since β > γ(1 − ε) + ε and α < γ(1 − ε) we have X_{β} ⊂ (X_{γ}, X_{1})_{ε} and (X_{γ}, X)_{ε} ⊂ X_{α}, respectively. Now the claim follows.

For the convenience of the reader we include the following Lemma from [18], which also gives a simpler proof of Corollary 6 of [17] (or [9] Section 4.1.). The original proof was based on interpolation theory.

\underline{3.5. Lemma}: Let a > ω(T_{t}). Then the functions

α ∈ (0, ∞) → w_{α}(x) = w((a − A)^{−α}x), x ∈ X fixed
α ∈ (0, ∞) → w_{α}(A) := sup{w((a − A)^{−α}x) : x ∈ U_{x}}

are convex and therefore continuous. \underline{Proof:} Put B^{α} = (a − A)^{−α}. Let 0 < α < β and γ = (1 − θ)α + θβ for some 0 < θ < 1. For u > w_{α}(x), v > w_{β}(x), choose a constant M such that ∥T_{t}(B^{α}x)∥ ≤ Me^{ut}, ∥T_{t}(B^{β}x)∥ ≤ Me^{vt}. Now we apply the convexity estimate (see [5], theorem 8.1)

∥B^{−c}y∥ ≤ ∥B^{−a}y∥^{1−θ}∥B^{−b}y∥^{θ}, a, b > 0, c = (1 − θ)a + θb

for y ∈ D(B^{−1}) to y = BT_{t}x and a = 1 − α, b = 1 − β, c = 1 − γ. Since T_{t} and B^{α} commute we find

∥T_{t}B^{γ}x∥ ≤ C∥B^{α}T_{t}x∥^{1−θ}∥B^{β}T_{t}x∥^{θ}
≤ MCe^{u(1−θ)t}e^{vθt}.

Hence w_{γ}(x) ≤ (1 − θ)w_{α}(x) + θw_{β}(x). The function w_{α}(A) is just the supremum of a set of convex functions.

\underline{3.6. Remark:} Lemma 3.5 also provides a partial answer to question on page of [9]. Indeed, the result of [18] or Corollary 4.3. of [9] and the continuity of α → ω(R(a, A)^{α}x) imply the following result: Let X be a Banach space of Fourier–type 1 ≤ p ≤ 2. If x ∈ X_{\frac{1}{p}} and the local resolvent λ → R(λ, A)x admits a bounded holomorphic extention F (λ) to the open right half–plane, then ω(x) ≤ 0.

\underline{3.7. Remark:} The spaces Bp,1^{\frac{n}{p}}, (IR^{n}, X) form a scale since by [13], Remark 6,

Bp,1^{\frac{n}{p}}(IR^{n}, X) ⊂ Bq,1^{\frac{n}{q}}(IR^{n}, X) if p < q.

The assumption of Theorem 2.1 is sharp in the sense, that the Theorem does not hold anymore for all Banach spaces X and Y of Fourier–type p, if we replace in its assumption the space B_{p,1}^{n/p}(Y) by the larger space B_{q,1}^{n/q}(Y) for some q > p. Otherwise one could prove that for every c_{0}–semigroup T_{t} with generator A on a Banach space X of Fourier–type p, one has

ω(x) ≤ s_{0}(A) for all x ∈ X_{α}, α ≥ \frac{1}{q} − \frac{1}{p′} .  (3.17)

(Indeed, the proof of Theorem 3.2 shows that for α ≥ \frac{1}{q} − \frac{1}{p′} , M(λ) = R(λi, A)^{2}

satisfies i) and ii) for the space W_{q}^{1}(X_{α}) in the place of W_{p}^{1}(X_{α}), and one could apply the multiplier Theorem to get (3.17)). But it was shown in [17] that

(3.17) fails for α < \frac{1}{p} − \frac{1}{p′}.

4 Stability of positive semigroups in L_{p}(μ).

Combing ideas from [7] and [16] we give a simple proof of the following theorem from [15]: \underline{4.1. Theorem:} If T_{t} is a positive semigroup on L_{p}(Ω, μ), 1 ≤ p < ∞, with generator A, then s(A) equals the growth bound of (T_{t}).

Assume s(A) < 0. According to the discussion in the beginning of Section 3, we only have to show that there is a constant C < ∞ such that for

K_{N}f(t) = ∫_{0}^{N} T_{s}(f(t − s))ds, f ∈ L_{p}(IR_{+}, L_{p}(Ω, μ))

we have

∥K_{N}f∥_{L_{p}(L_{p})} ≤ C∥f∥_{L_{p}(L_{p})} for all N.

For this we need the following result from [7]:

\underline{4.2. Lemma:} Let X and Y be Banach function spaces and P : X → Y a positive linear operator. If F : [0, N] → X is a Bochner–measurable function then

(∫ |P F (t)|^{p}dt)^{\frac{1}{p}} ≤ P (∫ |F (t)|^{p}dt)^{\frac{1}{p}} .

We will apply this lemma to X = L_{p}([0, N], L_{p}(μ)), Y = L_{p}(μ) and

P f = ∫_{0}^{N} T_{s}(f(s))ds,  for  f ∈ L_{p}([0, N], L_{p}(μ))

Define F : [0, N] → X by F (t)(s) = f(t−s)·χ_{IR_{+}}(t−s). Then by Fubini and 4.2

∥K_{N}f∥_{L_{p}(IR_{+},L_{p}(μ))} = ∥(∫ | ∫_{0}^{N} T_{s}f(t − s)ds|^{p}dt)^{\frac{1}{p}} ∥_{L_{p}(μ)}

= ∥(∫ |P F (t)|^{p}dt)^{\frac{1}{p}} ∥_{Y} ≤ ∥P (∫ |F (t)|^{p}dt)^{\frac{1}{p}} ∥_{Y}

= ∥ ∫_{0}^{N} T_{s}(∫ |f(t − s)|^{p}dt)^{\frac{1}{p}} ds∥ = ∥ ∫_{0}^{N} T_{s}gds∥
≤ ∥R(0, A)g∥ ≤ ∥R(0, A)∥ · ∥g∥

with g = (∫ |f(t)|^{p}dt)^{\frac{1}{p}} and ∥g∥_{L_{p}(μ)} = ∥f∥_{L_{p}(IR_{+},L_{p}(μ))}.

\underline{4.3 Remark:} For a positive semigroup we have that s_{0}(A) = s(A). It is therefore asked in ([9], question (iv) on page 147) if s_{0}(A) = ω(T_{t}) for all semigroups on L_{p}(μ)–spaces. This is indeed true for p = 2 according to a result of Gearhart (see e. g. [9]). But for p ≠ 2 the answer is negative, as shown by the following example from [16].

\underline{4.4 Example:} Let X = L_{p}(1, ∞)∩L_{2}(1, ∞) with norm ∥f∥ = ∥f∥_{2} +∥f∥_{p} for 2 < p < ∞. Consider the semigroup (S_{t}f)(x) = f(xe^{t}), t ≥ 0, with generator

(Bf)(x) = x(\frac{d}{dx}f)(x) on a suitable domain and (R(0, B)f)(x) = ∫_{x}^{∞}f(y)\frac{dy}{y}.

One can check that s(B) = −\frac{1}{2} < −\frac{1}{p} = ω(S_{t}) (cf. [2]). Since S_{t} is positive, we also have s_{0}(B) = s(B). By [6] 2.e.8(ii) and section 2.f. there is an isomorphism J of X onto L_{p}[0, 1] (essentially given by a stochastic integral with respect to the poisson process). Then the semigroup T_{t} = JS_{t}J^{−1} on L_{p}[0, 1] with generator A = JBJ^{−1} on D(A) = J(D(B)) still satisfies s_{∞}(A) = −\frac{1}{2} < −\frac{1}{p} = ω(T_{t}). If 1 < p < 2 we take the dual of T_{t} on L_{p′}, to obtain a similar example.

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Address: Lutz Weis, Universität Karlsruhe, Mathematisches Institut I, Englerstr. 2, D–76128 Karlsruhe, Germany. E–mail: Lutz.Weis@math.Uni–Karlsruhe.de

1991 Mathematics Subject Classification. Primary 47D06; Secondary 47N20